How to divide square roots - GRE Quantitative Reasoning

We then multiply our fraction by because we cannot leave a radical in the denominator. This gives us . Finally, we can simplify our fraction, dividing out a 3, leaving us with

We don't want to have radicals in the denominator. To get rid of radicals, just multiply top and bottom by that radical.

There are two methods we can use to simplify this fraction:

Method 1:

Factor the numerator:

Remember, we need to factor out perfect squares.

Method 2:

You can combine the fraction into one big square root.

Then, you can simplify the fraction.

Let's factor the square roots.

Then, multiply the numerator and the denominator by to get rid of the radical in the denominator.

We can definitely eliminate some answer choices. and don't make sense because we have an irrational number. Next, let's multiply the numerator and denominator of by . When we simplify radical fractions, we try to eliminate radicals, but here, we are going to go backwards.

, so is the answer.

We don't want to have radicals in the denominator. To get rid of radicals, just multiply the numerator and the denominator by that radical.

Remember to distribute the radical in the numerator when multiplying.

This may be the answer; however, the numerator can be simplified. Let's factor out the squares.

Finally, if we factor out a , we get:

Let's get rid of the radicals in the denominator of each individual fraction.

Then find the least common denominator of the fractions, which is , and multiply them so that they each have a denominator of .

We can definitely simplify the numerator in the right fraction by factoring out a perfect square of .

Finally, we can factor out a :

That's the final answer.

To get rid of the radical, we need to multiply by the conjugate. The conjugate uses the opposite sign and multiplying by it will let us rationalize the denominator in this problem. The goal is getting an expression of in which we are taking the differences of two squares.

This answer is the same as . Remember to distribute the negative sign.

To get rid of the radical, we need to multiply by the conjugate. The conjugate uses the opposite sign and multiplying by it will let us rationalize the denominator in this problem. The goal is getting an expression of in which we are taking the differences of two squares.

If we multiplied top and bottom by , we would get nowhere, as this would result: . Instead, let's cross-multiply.

Then, square both sides to get rid of the radical.

Divide both sides by .

The reason the negative is not an answer is because a negative value in a radical is an imaginary number.

To get rid of the radical, multiply top and bottom by .

Square both sides.

is "opposite over adjacent," or . (This is because .)

Let's substitute into the equation for and multiply the numerator and the denominator by .

Now we can multiply the result by :

is the inverse of , or :

We can reduce the resulting to :

By multiplying top and bottom by , we can cancel out the in the numerator:

The resulting fraction can be simplified:

We then multiply our fraction by because we cannot leave a radical in the denominator. This gives us . Finally, we can simplify our fraction, dividing out a 3, leaving us with

We don't want to have radicals in the denominator. To get rid of radicals, just multiply top and bottom by that radical.

There are two methods we can use to simplify this fraction:

Method 1:

Factor the numerator:

Remember, we need to factor out perfect squares.

Method 2:

You can combine the fraction into one big square root.

Then, you can simplify the fraction.

Let's factor the square roots.

Then, multiply the numerator and the denominator by to get rid of the radical in the denominator.

We can definitely eliminate some answer choices. and don't make sense because we have an irrational number. Next, let's multiply the numerator and denominator of by . When we simplify radical fractions, we try to eliminate radicals, but here, we are going to go backwards.

, so is the answer.

We don't want to have radicals in the denominator. To get rid of radicals, just multiply the numerator and the denominator by that radical.

Remember to distribute the radical in the numerator when multiplying.

This may be the answer; however, the numerator can be simplified. Let's factor out the squares.

Finally, if we factor out a , we get:

Let's get rid of the radicals in the denominator of each individual fraction.

Then find the least common denominator of the fractions, which is , and multiply them so that they each have a denominator of .

We can definitely simplify the numerator in the right fraction by factoring out a perfect square of .

Finally, we can factor out a :

That's the final answer.